Waves: An Interactive Tutorial Documents

Supplemental Documents (34)

This ePub document contains a preview of the Waves Tutorial. Use an ePub 3 reader that supports Math ML and JavaScript, such as the iBooks Reader on Apple devices or the Gitden on Android. The complete ePub tutorial is available in Apple iTunes.

Waves: An Interactive Tutorial is a set of 33 exercises designed to teach thefundamentals of wave dynamics. It starts with very simple wave properties and ends with anexamination of nonlinear wave behavior. The emphasis here is on the properties of waveswhich are difficult to illustrate in a static textbook figure. Simulations are not asubstitute for laboratory work. However they allow for visualization of processes thatcannot normally be seen (for example electric and magnetic fields). They allow forvisualization of process that are too fast (for example waves) to follow in real time or toosmall to see (for example thermodynamics at the molecular scale). They allow manipulation ofprocesses which might be dangerous (collisions) or hard to experiment with (waves). Theyalso allow for easy repetition. For all of these reasons, simulations are an excellent wayto introduce students to the complex phenomena of waves.

This simulation shows a perfect, smooth wave out on the ocean far enough from shore so that it has not started to break (complications involved in describing real waves will be discussed later in this tutorial).

Transverse waves are the kind of wave you usually think of when you think of a wave. The motion of the material constituting the wave is up and down so that as the wave moves forward the material moves perpendicular (or transverse) to the direction the wave moves.

The Longitudinal Waves simulation shows waves where the motion of the material is back and forth in the same direction that the wave moves. Sound waves (in air and in solids) are examples of longitudinal waves.

The Two-Dimensional Waves simulation shows a plane wave in two dimensions traveling in the x-y plane, in the x direction, viewed from above. In these simulations the amplitude (in the z direction, towards you) is represented in grey-scale.

Linear waves have the property, called superposition, that their amplitudes add linearly if they arrive at the same point at the same time. This simulation shows the sum of two wave functions u(x,t) = f(x,t) + g(x,t).

The Interference simulation shows a top view of a source making waves on the surface of a tank of water (imagine tapping the surface of a pond with the end of a stick at regular intervals). The white circles coming from the spot represents the wave crests with troughs in between. Two sources can be seen at the same time and the separation between them and the wavelength of both can be adjusted

The Group Velocity simulation shows how several waves add together to form a single wave shape (called the envelope) and how we can quantify the speed with two numbers, the group velocity of the combined wave and the phase velocity.

Fourier analysis is the process of mathematically breaking down a complex wave into a sum of of sines and cosines. Fourier synthesis is the process of building a particular wave shape by adding sines and cosines. Fourier analysis and synthesis can be done for any type of wave, not just the sound waves shown in this simulation.

This simulation shows how a standing wave is formed from two identical waves moving in opposite directions. For standing waves on a string the ends are fixed and there are nodes at the ends of the string. This limits the wavelengths that are possible which in turn determines the frequencies

This simulations represents a string as a row of individual masses connected by invisible springs. Waves are reflected in the middle of this string because the mass of the string is different on the left as compared with the right.

This simulation starts with the first four components of the Fourier series for a traveling square wave with no dispersion. Changing the angular frequency of a component causes the initial wave function to distort due to dispersion.

This simulation shows what happens to a plane-wave light source (below the simulation, not shown) as it passes through an opening. The wavelength of the waves and the size of the opening can be adjusted.

This simulation models at the Doppler effect for sound; the black circle is the source and the red circle is the receiver. If either the source or the receiver of a wave are in motion the apparent wavelength and frequency of the received wave change. This is apparent shift in frequency of a moving source or observer is called the Doppler Effect. The speed of the wave is not affected by the motion of the source or receiver and neither is the amplitude.

This simulation shows an accelerating positive charge and the electric field around it in two dimensions. Because the charge is accelerated there will be a disturbance in the field. The energy carried by the disturbance comes from the input energy needed to accelerate the charge.

This simulation shows the effect of a wave traveling in the x-direction on a second charge inside a receiving antenna. Only the y-component of the change in the electric field is shown (so an oscillation frequency of zero will show nothing, because there is only a constant electric field).

This simulation shows the electric field component[s] for a wave traveling straight towards the observer in the +y direction. A polarized wave was previously defined to be an electromagnetic wave that has its electric field confined to change in only one direction. In this simulation we further investigate polarized waves.

In this simulation we look at the dynamics of waves; the physical situations and laws give rise to waves. We start with a string that has a standing wave on it and look at the forces acting on each end of a small segment of the string due to the neighboring sections. For visualization purposes the string is shown as a series of masses but the physical system is a continuous string. Although the derivation is for a string, similar results occur in many other systems. The ends of the section of string we are interested in are marked with red dots in the simulation. The tension acting on each end is shown with a vector (in red) and its components (green and blue).

In this simulation we examine waves that occur on chains of masses with mass M coupled together with elastic, Hooke's law forces (F = -?x where ? is the spring constant and x is the amount the spring stretches). The masses are constrained to only move up and down so that the stretching depends only on the difference in the y locations of the masses.

This simulations shows what happens if forces other than tension act on a string. Some additional forces cause the dispersion we saw in simulations 22 and 23 but friction, dissipation and nonlinearity can cause other behavior as we will see here.

This simulation explores a special solution of the non-linear wave equation where the effects of dispersion and dissipation (which tend to make a wave pulse spread out) are exactly compensated for by a nonlinear force (which, as we have seen, tends to cause steepening of a wave). In this case there may be a special wave pulse shape that can travel and maintain its shape called a soliton.

Source Code Documents (2)

This source code zip archive contains an XML representation of the Sine Wave JavaScript Model. Unzip this archive in your Ejs workspace to compile and run this model using EjsS 5. Although EjsS is a Java program, EjsS 5 creates a stand alone JavaScript program from this source code.

This source code zip archive contains an XML representation of the Speed of a Wave JavaScript Model. Unzip this archive in your Ejs workspace to compile and run this model using EjsS 5. Although EjsS is a Java program, EjsS 5 creates a stand alone JavaScript program from this source code.